Equilibrium of Forces Acting at a Point
Click here1 for the revised instructions for this lab.
INTRODUCTION
Addition of Forces
Forces are one of a group of quantities known as vectors, which are distinguished from regular
numbers (known as scalars) by the fact that a vector has two quantities associated with it, a
magnitude and a direction (related to a coordinate axes of the system you are dealing with). These
properties completely characterize a vector.
A vector may alternatively be described by specifying its vector components. In the case of
the Cartesian coordinate system (the system we will be primarily dealing with) there are two
components, the x-component and y-component. These two properties also completely characterize
a vector. Vectors, and in the case of this lab, force vectors, can be represented pictorially (see Fig.
1) by an arrow pointing in the direction of action of the force, with a length proportional to the
strength (magnitude) of the force.
Figure 1
The components Fx and Fy in the x and y directions of the vector F are related to the magnitude
F and angle θ by:
Fx = F cos θ and Fy = F sin θ
(1)
and conversely:
q
Fy
2
2
F = Fx + Fy , and θ = arctan
.
Fx
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(2)
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When several forces act on a point, their sum can be obtained according to the rules of vector
algebra. Graphically, the sum of two forces F = F1 + F2 can be found by using the parallelogram
rule illustrated in Fig. 2 or, equivalently, by the head-to-tail method illustrated in Fig. 3.
Figure 2
Figure 3
The sum of the vectors can also be derived analytically by adding their components:
Fx = F1x + F2x , and Fy = F1y + F2y
(3)
Condition for Translational Equilibrium
An object is in translational equilibrium when the vector sum of all the forces acting on it is
zero. In this experiment we shall study the translational equilibrium of a small ring acted on by
several forces on an apparatus known as a force table, see Fig. 4. This apparatus enables one
to cause the forces of gravity acting on several masses (F = mg) to be brought to bear on the
small ring. These forces are adjusted until equilibrium of the ring is achieved. You will then add
the forces analytically by adding their components and graphically by drawing the vectors and
determining if they add to zero using the rules for the addition of force vectors listed above.
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2011
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Figure 4
PROCEDURE
Equilibrium with Three Forces
We shall first study the equilibrium of the small ring when there are three forces acting on it.
Two of the forces (F1 and F2 ) will be fixed and the third one F3 adjusted until equilibrium is
reached.
1
If necessary, level the force table using the small bubble level placed on the table’s surface.
2
Choose any two masses you like in the range 100-300 g, and place each mass on a weight holder.
Use the electronic balance to measure each of the masses including the holder. Designate the
measured masses as m1 and m2 . The uncertainty of these measurements should be limited to
the precision of the balance.
3
Place the pin in the middle of the force table and place the ring over the pin. Attach two of
the four pulleys provided to the force table at any position other than zero degrees. Record the
value of θ1 and θ2 . The uncertainty in these angles should be limited to the precision to which
you can read the angles on the force table.
4
Run two of the strings (attached to the ring) over the pulleys, and suspend the masses that
you have chosen at the appropriate angles (m1 at θ1 and m2 at θ2 ). The tension in the two
strings acts on the ring with forces F1 and F2 , each with a magnitude equal to the weight of the
corresponding mass and holder (m1 g and m2 g) suspended at the end of each of the strings.
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Pull one of the remaining strings in various directions until you locate a direction in which the
ring is freed from the pin when you apply the right amount of force. Attach a third pulley at this
position. Run the string over the pulley and attach a weight holder to the string. Add weights
to the weight holder until the ring pulls away from the pin, so that the pin is not necessary
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to hold the ring in place. This last added force is the (equilibriant) force F3 (m3 g). It may
be necessary to make minor adjustments to the angle to obtain a precise measurement. Make
sure that the strings are stretched radially and the pin is at the center of the ring. Estimate
the uncertainty in the equilibrant force by adjusting the mass and angle until the system is no
longer in equilibrium.
Equilibrium with Four Forces
1
Now select three masses (to provide three forces with sum F1 + F2 + F3 ) at three angles (one
of them zero) and determine what fourth single mass and angle establishes equilibrium on the
force table (the equilibriant force F4 ).
2
Record all angles, masses and their uncertainties as in Equilibrium with Three Forces.
Be sure to pledge your work, initial your data, and have your TA initial your
data.
ANALYSIS
Graphical Analysis
Make accurate diagrams on normal rectangular graph paper showing the sum of the forces acting
on the ring for both parts of the experiment above.
1
Draw force diagrams to scale. For example, 5 Newtons = 1 cm. Use whatever works best to
give you the greatest plotting precision.
2
Use the head-to-tail method to find the sum of the forces graphically. Be as accurate as possible.
Qualitatively verify that the sum is zero. If it is not, determine from your graph the magnitude
of the deviation from zero.
Analytical Sum
Calculate the resultant force on the ring, FT = F1 + F2 + F3 , analytically for Equilibrium
with Three Forces only. Choose zero degree to be the +x -axis, and 90◦ to be the +y-axis.
The Analytical Sum section of the WebAssign question for this lab, will help facilitate the error
analysis.
1
Use the tables in the WebAssign question to enter the data for the forces acting on the ring.
For each force, include the magnitude F, its uncertainty u F , the direction θ, and its uncertainty
uθ . The values of uθ must be expressed in units of radians.
2
Calculate the x - and the y-components of each of the forces together with their errors. Pay
attention to the sign of each component. Include in the last row of the table the sum of
the components and their error.
3
Calculate the magnitude and the direction of the resultant force FT . Compute also the uncertainty of the magnitude.
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DISCUSSION
Is the condition for static equilibrium, FT = 0, satisfied for both parts of the experiment?
How does your uncertainty of FT compare to the precision of your force and angle measurements?
Discuss the sources of systematic error and how they affect your results. What is the primary
source of error in this experiment? Discuss attempts you have made to reduce both systematic and
random errors. What did you learn or discover from this lab? When might you apply the skills
learned from this lab?
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